IIT Maths Sample Paper 1

Algebra

1. Let x be a real number with 0<x<p. Prove that, for all natural numbers n, the sum sinx + sin3x/3 + sin5x/5 + ... + sin(2n-1)x/(2n-1) is positive.
2. Use combinatorial argument to prove the identity:
   _¥
   å   ^n-r+1C_d . ^r-1C_d-1 = ⁿC_r
   ^d=1
3. a, b are roots of x²+ax+b=0, g, d are roots of x²-ax+b-2=0. Given 1/a + 1/b4 + 1/d + 1/g =5/12 and abdg = 24, find the value of the coefficient ?a?.
4. x + y + z = 15 and xy + yz + zx = 72, prove that 3 £ x £ 7.
5. Let l, a Î R, find all the set of all values of l for which the set of linear equations has a non-trivial solution.
   lx + (sin a) y + (cos a) z = 0
   x + (cos a) y + (sin a) z = 0
   -x + (sin a) y - (cos a) z = 0
   If l = 1, find all values of a.
6. Prove that for each posive integer 'm' the smallest integer which exceeds (Ö3 + 1)^2m is divisible by 2^m+1.
7. Prove that, for every natural k, the number (k³)! is divisible by (k!)^k2+k+1.
8. Prove that the inequality: _n=1å^r  { _m=1å^r  a_ma_n/(m+n)} ³ 0. a_i is any real number.
9. Prove the following inequality: _k=1åⁿ   Ö[ⁿC_k] £ Ö[n(2ⁿ-1)]
10. A sequence {U_n, n ³ 0} is defined by U₀=U₁=1 and U_n+2=U_n+1+U_n.Let A and B be natural numbers such that A¹⁹ divides B⁹³ and B¹⁹ divides A⁹³.Prove by mathematical induction, or otherwise, that the number (A⁴+B⁸)^Un+1 is divisible by (AB)^Un for n ³ 1.
11. The real numbers a, b satisfy the equations: a³ + 3a² + 5a - 17 = 0, b³ - 3b² + 5b + 11 = 0. Find a+b.
12. Given 6 numbers which satisfy the relations:
    y² + yz + z² = a²
    z² + zx + x² = b²
    x² + xy + y² = c²
    Determine the sum x+y+z in terms of a, b, c. Give geometrical interpretation if the numbers are all positive.
13. Solve: 4x²/{1-Ö(1+2x²)}² < 2x+9
14. Find all real roots of: Ö(x²-p) + 2Ö(x²-1) = x
15. The solutions a, b, g of the equation x³+ax+a=0, where 'a' is real and a¹0, satisfy a²/b + b²/g + g²/a = -8. Find a, b, g.
16. If a, b, c are real numbers such that a²+b²+c²=1, prove the inequalities: -1/2 £ ab+bc+ca £ 1.
17. Show that, if the real numbers a, b, c, A, B, C satisfy: aC-2bB+cA=0 and ac-b²>0 then AC-B²£0.
18. When 0<x<1, find the sum of the infinite series: 1/(1-x)(1-x³) + x²/(1-x³)(1-x⁵) + x⁴/(1-x⁵)(1-x⁷) + ....
19. Solve for x, y, z:
    yz = a(y+z) + r
    zx = a(z+x) + s
    xy = a(x+y) + t
20. Solve for x, n, r > 1
    

    xCr n-1Cr n-1Cr-1 x+1Cr nCr nCr-1 x+2Cr n+2Cr n+2Cr-1

    = 0

21. Let p be a prime and m a positive integer. By mathematical induction on m, or otherwise, prove that whenever r is an integer such that p does not divide r, p divides ^mpC_r.
22. Let a and b be real numbers for which the equation x⁴ + ax³ + bx² + ax + 1 = 0 has at least 1 real solution. For all such pairs (a,b), find the minimum value of a²+b².
23. Prove that:
    2/(x² - 1) + 4/(x² - 4) + 6/(x² - 9) + ... + 20/(x² - 100) =
      11/((x - 1)(x + 10)) + 11/((x - 2)(x + 9)) + ... + 11/((x - 10)(x + 1))
24. Find all real p, q, a, b such that we have (2x-1)²⁰ - (ax+b)²⁰ = (x²+px+q)¹⁰ for all x.